UNIVERSITÉ DE MONTRÉAL ARC ROUTING PROBLEMS FOR ROAD NETWORK MAINTENANCE INGRID MARCELA MONROY LICHT

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1 UNIVERSITÉ DE MONTRÉAL ARC ROUTING PROBLEMS FOR ROAD NETWORK MAINTENANCE INGRID MARCELA MONROY LICHT DÉPARTEMENT DE MATHÉMATIQUES ET DE GÉNIE INDUSTRIEL ÉCOLE POLYTECHNIQUE DE MONTRÉAL THÈSE PRÉSENTÉE EN VUE DE L OBTENTION DU DIPLÔME DE PHILOSOPHIÆ DOCTOR (GÉNIE INDUSTRIEL) AOÛT 2015 Ingrid Marcela Monroy Licht, 2015.

2 UNIVERSITÉ DE MONTRÉAL ÉCOLE POLYTECHNIQUE DE MONTRÉAL Cette thèse intitulée : ARC ROUTING PROBLEMS FOR ROAD NETWORK MAINTENANCE présentée par : MONROY LICHT Ingrid Marcela en vue de l obtention du diplôme de : Philosophiæ Doctor a été dûment acceptée par le jury d examen constitué de : M. GENDREAU Michel, Ph. D., président M. LANGEVIN André, Ph. D., membre et directeur de recherche M. AMAYA GUIO, Ciro Alberto, Ph. D., membre et codirecteur de recherche M. ROUSSEAU Louis-Martin, Ph. D., membre et codirecteur de recherche M. GAMACHE Michel, Ph. D., membre M. EGLESE Richard, Ph. D., membre externe

3 iii DEDICATION To my Parents Astrid and Fernando, To my Grandmother Cecilia

4 iv ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my supervisor Dr. André Langevin for his valuable guidance and support. He was always available and patient for my questions. I owe him my gratitude for supporting me in difficult moments. I would also like to thank my supervisor Dr. Ciro Alberto Amaya for sharing expertise, valuable guidance and encouragement extended to me. I am also extremely grateful to my supervisor Dr. Louis-Martin Rousseau for his help, valuable guidelines and suggestions. I wish to thank the jury members of my dissertation for taking the time to read this work. I owe my thanks to the staff at Department of Mathematics and Industrial Engineering and at Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT). Finally, I would like to thank my family for their support and for being there for me through ups and downs.

5 v RÉSUMÉ Cette thèse présente deux problèmes rencontrés dans l entretien des réseaux routiers, soit la surveillance des réseaux routiers pour la détection de verglas sur la chaussée et la reprogrammation des itinéraires pour les activités de déneigement et d épandage de sel. Nous représentons ces problèmes par des modèles de tournées sur les arcs. La dépendance aux moments et la nature dynamique sont des caractéristiques propres de ces problèmes, par conséquence le cas de surveillance des réseaux routiers est modélisé comme un problème de postier rural avec fenêtres-horaires (RPPTW), tandis que le cas de la reprogrammation utilise des modèles obtenus à partir des formulations de problèmes de tournées sur les arcs avec capacité. Dans le cas du problème de surveillance, une patrouille vérifie l état des chemins et des autoroutes, elle doit principalement détecter le verglas sur la chaussée dans le but d assurer de bonnes conditions aux chauffeurs et aux piétons. Étant donné un réseau routier et des prévisions météo, le problème consiste à créer une tournée qui permette de détecter opportunément le verglas sur les rues et les routes. L objectif poursuivi consiste à minimiser le coût de cette opération. En premier, on présente trois formulations basées sur la programmation linéaire en nombres entiers pour le problème de surveillance des réseaux qui dépend du moment et deux méthodes de résolution: un algorithme de coupes et un algorithme heuristique appelé adaptive large neighborhood search (ALNS). La méthode exacte inclut des inéquations valides tirées du problème du voyageur de commerce avec fenêtres-horaires et aussi du problème de voyageur du commerce avec contraintes de précédence. La méthode heuristique considère deux phases: en premier, on trouve une solution initiale et après dans la deuxième phase, l algorithme essaie d améliorer la solution initiale en utilisant sept heuristiques de destruction et deux heuristiques de réparation choisies au hasard. La performance des heuristiques est évaluée pendant les itérations. Une meilleure performance correspond à une plus grande probabilité de choisir une heuristique. Plusieurs tests ont été faits sur deux ensembles d exemplaires de problèmes. Les résultats obtenus montrent que l algorithme de coupes est capable de résoudre des réseaux avec 104 arêtes requises et des fenêtres-horaires structurées par tranches horaires ; l algorithme peut aussi

6 vi résoudre des réseaux avec 45 arêtes requises et des fenêtres-horaires structurées pour chaque arête requise. Pour l algorithme ALNS, différentes versions de l algorithme sont comparées. Les résultats montrent que cette méthode est efficace parce qu elle est capable de résoudre à l optimalité 224 des 232 exemplaires et de réduire le temps de calcul significativement pour les exemplaires les plus difficiles. La dernière partie de la thèse introduit le problème de la reprogrammation de tournées sur les arcs avec capacité (RCARP), lequel permet de modéliser la reprogrammation des itinéraires après une panne d un véhicule lors de la phase d exécution d un plan initial des activités de déneigement ou d épandage de sel. Le planificateur doit alors modifier le plan initial rapidement et reprogrammer les véhicules qui restent pour finir les activités. Dans ce cas, l objectif poursuivi consiste à minimiser le coût d opération et le coût de perturbation. La distance couverte par les véhicules correspond au coût d opération, cependant une nouvelle métrique est développée pour mesurer le coût de perturbation. Les coûts considérés sont des objectifs en conflit. On analyse quatre politiques à la phase de re-routage en utilisant des formulations de programmation linéaire en nombres entiers. On propose une solution heuristique comme méthode pour résoudre le RCARP quand les coûts d opération et de perturbation sont minimisés en même temps et quand une réponse rapide est nécessaire. La méthode consiste à fixer une partie de l itinéraire initial et après à modifier seulement les itinéraires des véhicules les plus proches de la zone de l interruption de la tournée du véhicule défaillant. La méthode a été testée sur des exemplaires obtenus d un réseau réel. Nos tests indiquent que la méthode peut résoudre rapidement des exemplaires avec 88 arêtes requises et 10 véhicules actifs après la panne d un véhicule. En conclusion, la principale contribution de cette thèse est de présenter des modèles de tournées sur les arcs et de proposer des méthodes de résolution d optimisation qui incluent la dépendance aux temps et l aspect dynamique. On propose des modèles et des méthodes pour résoudre le RPPTW, et on présente des résultats pour ce problème. On introduit pour la première fois le RCARP. Trois articles correspondant aux trois principaux chapitres ont été acceptés ou soumis à des revues avec comité de lecture: The rural postman problem with time windows accepté dans Networks, ALNS for the rural postman problem with time windows soumis à Networks, and

7 vii The rescheduling capacitated arc routing problem soumis à International Transactions in Operational Research.

8 viii ABSTRACT This dissertation addresses two problems related to road network maintenance: the road network monitoring of black-ice and the rescheduling of itineraries for snow plowing and salt spreading operations. These problems can naturally be represented using arc routing models. Timing-sensitive and dynamic nature are inherent characteristics of these problems, therefore the road network monitoring is modeled as a rural postman problem with time windows (RPPTW) and in the rescheduling case, models based on capacitated arc routing formulations are suggested for the rerouting phase. The detection of black-ice on the roads is carried out by a patrol to ensure safety conditions for drivers and pedestrians. Specific meteorological conditions cause black-ice on the roads; therefore the patrol must design a route covering part of the network in order to timely detect the black-ice according to weather forecasts. We look for minimum-cost solutions that satisfy the timing constraints. At first, three formulations based on mixed integer linear programming are presented for the timing-sensitive road network monitoring and two solution approaches are proposed: a cutting plane algorithm and an adaptive large neighborhood search (ALNS) algorithm. The exact method includes valid inequalities from the traveling salesman problem (TSP) with time windows and from the precedence constrained TSP. The heuristic method consists of two phases: an initial solution is obtained, and then in the second phase the ALNS method tries to improve the initial solution using seven removal and two insertion heuristics. The performance of the heuristics is evaluated during the iterations, and therefore the heuristics are selected depending on their performance (with higher probability for the better ones). Several tests are done on two sets of instances. The computational experiments performed show that the cutting plane algorithm is able to solve instances with up to 104 required edges and with time windows structured by time slots, and problems with up to 45 required edges and time windows structured by each required edge. For the ALNS algorithm, several versions of the algorithm are compared. The results show that this approach is efficient, solving to optimality 224 of 232 instances and significantly reducing the computational time on the hardest instances.

9 ix The last part of the dissertation introduces the rescheduling capacitated arc routing problem (RCARP), which models the rescheduling of itineraries after a vehicle failure happens in the execution of an initial plan of snow plowing or salt spreading operations. A dispatcher must quickly adjust the remaining vehicles and modify the initial plan in order to complete the operations. In this case we look for solutions that minimize operational and disruption costs. The traveled distance represents the operational cost, and a new metric is discussed as disruption cost. The concerned objectives are in conflict. Four policies are analyzed in the rerouting phase using mixed integer linear programming formulations. A heuristic solution is developed to solve the RCARP when operational and disruption costs are minimized simultaneously and a quick response is needed. The idea is to fix part of the initial itinerary and only modify the itinerary of vehicles closer to the failure zone. The method is tested on a set of instances generated from a real network. Our tests indicate that the method can solve instances with up to 88 required edges and 10 active vehicles after the vehicle breakdown. In short the main contribution of this dissertation is to present arc routing models and optimization solution techniques that consider timing-sensitive and dynamic aspects. Formulations and solution methods with computational results are given for the RPPTW, and the RCARP is studied for the first time here. Three articles corresponding to the main three chapters have been accepted or submitted to peer review journals: The rural postman problem with time windows accepted in Networks, ALNS for the rural postman problem with time windows submitted to Networks, and The rescheduling capacitated arc routing problem submitted to International Transactions in Operational Research.

10 x TABLE OF CONTENTS DEDICATION... III ACKNOWLEDGEMENTS... IV RÉSUMÉ... V ABSTRACT... VIII TABLE OF CONTENTS... X LIST OF TABLES... XIV LIST OF FIGURES... XV LIST OF SYMBOLS AND ABBREVIATIONS... XVI CHAPTER 1 INTRODUCTION Thesis Outline... 3 CHAPTER 2 LITERATURE REVIEW Arc routing problems with time windows The Chinese postman problem The rural postman problem Capacitated arc routing problem Dynamic arc routing problems Dynamic rural postman problem Dynamic CARP CHAPTER 3 ARTICLE 1 : THE RURAL POSTMAN PROBLEM WITH TIME WINDOWS 28 Abstract Introduction... 28

11 xi 3.2 Undirected RPPTW Model on the edges Model on the required edges Model on the nodes Valid inequalities Solution algorithm Data preprocessing Cutting plane algorithm Solution of the MIP program Computational results Generated instances Instances based on the Estrie network Preprocessing Tests Directed case Model on the arcs Model on the required arcs Tests Conclusions References CHAPTER 4 ARTICLE 2 : ALNS FOR THE RURAL POSTMAN PROBLEM WITH TIME WINDOWS 56 Abstract Introduction Literature review... 57

12 xii 4.3 Adaptive large neighborhood search Initial solution Improvement phase Results Instances Tuning set parameters Performance of removal and insertion heuristics Results for set Summary of computational results Conclusions References CHAPTER 5 ARTICLE 3 : THE RESCHEDULING CAPACITATED ARC ROUTING PROBLEM.79 Abstract Introduction Problem definition Measures of disruption cost Edit distance Exact match R-type distance Longest sequence Formulations Objective 1 (O1): Minimizing the total distance traveled Objective 2 (O2): Minimizing the total distance traveled and considering capacity Objective 3 (O3): Minimizing disruption cost... 87

13 xiii Objective 4 (O4): Minimizing operational and disruption cost Results Test set and baseline solution Comparison of policies Larger networks Solution strategy Evaluation of metrics Conclusions References Annex 1. Solution strategy CHAPTER 6 GENERAL DISCUSSION CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS REFERENCES

14 xiv LIST OF TABLES Table 2.1: Synthesis of works in road network maintenance considering time-sensitive and dynamic context... 8 Table 3.1: Costs of the transformed graph Model on the required edges Table 3.2: Costs of the transformed graph Model on the nodes Table 3.3: Reduction in number of variables Table 3.4: Models comparison and cutting plane algorithm Table 3.5: Cutting plane on real instances Table 3.6: Model on the arcs set of instances setd Table 4.1: Optimal solutions for benchmark instances Table 4.2: Performance of single versions of the ALNS on set Table 4.3: Gaps for single versions of the ALNS on set Table 4.4: Performance of versions VR124I2 and VR124I12 on set Table 4.5: Gaps for VR124I2 and VR124I12 on set Table 4.6: Comparison of VR124I12 and cutting plane algorithm Table 4.7: Average gap: Study of the effect of q and time limit Table 4.8. Best solutions: set Table 5.1: Characteristic of problems Table 5.2: Comparison of objectives Table 5.3: Comparison of results for larger networks Table 5.4: Computational times (Problem 12_2, BB = 0.5tt) Table 5.5: Strategy solution in larger instances

15 xv LIST OF FIGURES Figure 2.1: Types of time windows Figure 2.2: Example of time windows in an undirected graph Figure 3.1: Transformed graph for the model on the required edges Figure 3.2: Transformed graph for the model on the nodes Figure 3.3: Estrie network Weather forecast for one time slot Figure 3.4: Transformed graph for the model on the required arcs Figure 5.1: Comparison of travel and disruption costs Figure 5.2: Travel and disruption costs (Problem 12_2, BB = 0.5tt) Figure 5.3: Example of different policies for problem 12_2, BB =0.5tt Figure 5.4: Disruption metrics

16 xvi LIST OF SYMBOLS AND ABBREVIATIONS ALNS ATSP-TW CARP CARPTW CPP CPPTW GRASP LP MIP PC-ATSP RCARP RWIS RPP RPPTW TSP VND VRP VRPTW Adaptive Large Neighborhood Search Asymmetric Traveling Salesman Problem with Time Windows Capacitated Arc Routing Problem Capacitated Arc Routing Problem with Time Windows Chinese Postman Problem Chinese Postman Problem with Time Windows Greedy Randomized Adaptive Search Procedure Linear Program Mixed integer Program Precedence Constrained Asymmetric Traveling Salesman Problem Rescheduling Capacitated Arc Routing Problem Road Weather Information System Rural Postman Problem Rural Postman Problem with Time Windows Traveling Salesman Problem Variable Neighborhood Descend Vehicle Routing Problem Vehicle Routing Problem with Time Windows

17 1 CHAPTER 1 INTRODUCTION Road network maintenance, especially in winter is a significant challenge to most governments and transportation agencies in North America. An indication of the magnitude of this operation is related to the high costs that it implies. In Ontario, the total expenditures in highway winter maintenance reached $171 million in the 2013 fiscal year (Office of the Auditor General of Ontario, 2015). Michigan Department of Transportation spent $103 million for fiscal year 2013 (Slone, 2014). The New Jersey Department of Transportation reported that it spent a record $138 million to keep state roadways clear of snow and ice for 2013 (Slone, 2014). The Pennsylvania Department of Transportation, which had $189.2 million budgeted for the winter, spent $284 million (Slone, 2014). Edmonton, the city which spends more on snow removal than any other city in western Canada, estimated a budget of $50.4 for all the winter road maintenance operations for 2013 (Rodrigues, 2013). Bob Dunford, Director of Roadway Maintenance in Edmonton, listed the expenses of winter maintenance in the city (interview given to Rodrigues (2013)): The plowing we do is about $19 million and sanding is about $11 million, removal is about $6.1 million and then you ve got a snow storage site at about $2 million. Also, we have 1100 km of sidewalks that we are responsible for, and that s $7 million a year we spend on that. Winter road maintenance includes all the operations that aim at the removal or reduction of snow and ice on roadways providing safe winter driving conditions and safe sidewalks for pedestrians. The roads and highways must be kept cleared of snow and ice on a reasonably time basis. Providing an efficient winter road service is a responsibility of municipalities, but in many cases they outsource the winter maintenance operations to private-sector contractors. Winter maintenance authorities are constantly seeking for technology-based solutions such as advanced road weather information systems for monitoring localized weather and road surface conditions, automated vehicle location and Global Positioning System for tracking fleet operations and performance (Fu et al. 2009). However to obtain a great benefit from these technologies, the integration with algorithms that support the decisions and plans in operational, strategic and tactical level of road winter maintenance is necessary. Decision-making at the supervisory level of winter maintenance operations are often complex and constrained by time

18 2 and resources. For example for salt spreading, Eglese (1994) stated that if a road is treated too early, then the salt may be washed away if rain is falling or blown away by the wind before the temperature drops to freezing conditions. If the road is treated too late, then ice may have already started to form and the road will be dangerous for traffic traveling on the road before it has been treated. Arc routing is an operational research area that has contributed with models and algorithms to assist road network maintenance managers and operators to make more structured decisions. The use of operational research in this field could results in substantial saving, improved mobility, and reduced environmental and social impacts (Liu et al. 2014). However, for many years the focus of arc routing in winter maintenance has been in static problems where all data are assumed known before the routes are constructed and do not change afterward. But the reality of winter road maintenance is really complex and has a dynamic nature. The new information technologies mentioned above, and other technological advances in communication systems allow the exploration of real time information for dynamic routing and scheduling. This favors getting more realistic representation of the operations and enhancing the performance of decision systems in the area of routing. Few works have attempted to address the time-sensitive and dynamic nature of winter maintenance operations. This Dissertation focuses on these two complex issues of road network maintenance. The objective is to develop decision support arc routing models and optimization solution methods for two cases: road network monitoring and rescheduling of salt spreading and snow plowing operations. The first case considers the time sensitivity due to weather forecasts in the schedule of the monitoring of black-ice on roads, and the second case considers a dynamic component, where given an initial schedule for spreading or plowing operations, it must be modified when a vehicle breakdown occurs. Road network monitoring: In the region of Estrie in the Province of Quebec, from mid- October to mid-december a patrol daily checks the state of roads and must detect black ice on roads in a timely fashion in order to avoid pedestrian falls or automobile accidents. Once the patrol has noticed a possible risk for safe mobility, it reports the incident to service centers in charge of road signs and special works. The planner of the route has access to short-term weather forecasts and a characterization of roads with likelihood of black-ice formation. The planner must

19 3 match the temperature conditions in different zones with the probability of the presence of blackice and plan a circuit for the patrol. Rescheduling of salt spreading - snow plowing operations: Snow removal and salt spreading, the most common activities in winter maintenance, are often performed for each storm event on a repetitive basis over the same routes until safe mobility conditions are achieved (Campbell, Langevin, and Perrier, 2014). In practice master schedules are usually constructed by solving deterministic capacitated arc routing instances based on average demands predicted by the weather conditions. The master schedules should be executed as they were established if no unexpected events occur. However, disruptions may occur, especially in winter, which may interrupt the master plans. When a disruption occurs, routes should be quickly revised to minimise the negative impact it may cause and to ensure the quality of service. The case of adjusting an initial itinerary after one or more vehicle breakdowns during the execution stage is considered in this research. This dissertation consists of four main chapters: Chapter 2 presents the literature review. In Chapter 3 an exact approach is proposed to solve the rural postman problem with time windows (RPPTW); the article of Chapter 3 has been accepted for publication in Networks. In Chapter 4 a heuristic method is presented to solve larger instances of the same problem; the corresponding article has been submitted to Networks. In Chapter 5 the rescheduling capacitated arc routing problem is introduced considering operational and disruption costs; the article of Chapter 5 has been submitted to International Transactions in Operational Research. A more detailed outline of the thesis is given in the following section. 1.1 Thesis Outline At first, the contributions in the literature on time-sensitive and dynamic problems in arc routing problems, especially in road winter maintenance applications are reviewed. This review concludes that more research is needed on time-sensitive and dynamic winter maintenance arc routing problems. Then, Chapters 3 and 4 present the case of road network monitoring. In the first article (Chapter 3) the monitoring of roads for black-ice detection is modeled as a RPPTW. In the literature it is known that the polyhedral theory is very successful in solving postman problems,

20 4 especially the NP-hard postman problems (Tan et al. 2013). This approach is explored after presenting several mathematical formulations of the problem. A cutting plane algorithm is proposed as a solution method and tested on sets of instances adapted from the literature and on the real network of the Estrie region. The second article (Chapter 4) presents a metaheuristic as a solution technique to the monitoring of road network when large instances are considered. The technique is based on the adaptive large neighborhood search metaheuristic, which is one of the most successful metaheuristics for solving complex routing problems (Ropke and Pisinger, 2006a). Several versions of the metaheuristic combining different removal operators are compared. The performance of the metaheuristics is evaluated in comparison to the solution of the first approach. Regarding to dynamic issues in road winter maintenance operations, the third article (Chapter 5) deals with the vehicle failures during the execution phase of salt spreading or snow plowing operations. In dynamic routing problems all or part of the information are revealed or updated as the routes are executed. Hence, the planners must react to events that occur in real time. This work considers a reschedule as response to the disruption in a capacitated arc routing problem itinerary. The objective is to study different policies in the rerouting phase considering operational and disruption costs. Formulations based on mixed integer programming are presented and a solution strategy is discussed when operational and disruption costs are minimized simultaneously.

21 5 CHAPTER 2 LITERATURE REVIEW An important area for the applications of arc routing is concerned with road network maintenance. In particular, there are various logistic operations that deal with maintenance of roads in winter. Spreading chemicals and abrasives, plowing roadways and sidewalks, loading snow into trucks, and ice control are some examples of operations which are key activities to maintaining the safety and the mobility in cities and rural areas. The complex operations, the infrastructure constraints, especially in urban areas, and the dynamic nature of the operating conditions make the road network maintenance a challenge for many governments. Arc routing problems in the context of road network maintenance differ from other arc routing applications in a number of important ways due to climate, level of service, network complexity and size, traffic conditions, turn restrictions, synchronization of operations, policy decisions and others. Therefore, arc routing models and solutions that consider practical complexities of the problem are of enormous worth. Campbell and Langevin (2000) present a brief history of roadway snow and ice control in the U.S. from 1862 to 1996 and a survey of early works addressing arc routing in the same frame from 1970 to In addition, a general description of two successful software packages for roadway snow and ice control is presented. The first one CASPER, developed in India incorporates multiple objectives in designing routes via a penalty function. The objectives included in the software consider: meeting specific service level (time limit for routes), minimizing deadhead travel, and maintaining class continuity when priorities are given. The second software package is GeoRoute Municipal, which has been developed in Canada; its module route manager is its optimization system. It works dividing the region of study into sectors, and then for each to optimize the routes for a number of scenarios. The survey provides evidence of the wide gap between theory and practice, therefore authors encourage contributions in the area of route optimization for winter road maintenance. Perrier et al. (2006a, 2006b, 2007a, 2007b) present a comprehensive review of models and algorithms developed for the variety of winter road maintenance operations. This work is divided into 4 surveys. The first one focuses on optimization models and solution algorithms for the

22 6 design for spreading and plowing. The second one discusses system design problems for snow disposal operations. The last two address vehicle routing, depot location, and fleet sizing models for winter road maintenance. Perrier et al. (2012) provide a survey of recent optimization models and solution methodologies for the routing of spreading operations. They present a detailed classification scheme for spreader routing models developed over the past 40 years. They emphasize that the new models demonstrate impressive capacities to include more issues of the real complexity, the use of more sophisticated hybrid solutions strategies and consideration of more comprehensive models that integrate vehicle routing with other strategic winter maintenance problems. However, they note that there is still a large gap between state-of-the-art models and actual implementations. Campbell et al. (2014) present the most recent survey on operational research methods on snow plow routing and a case study on implementation of route optimization for snow plowing. They report the works not covered in the survey of Perrier et al. (2007b). This review documented the trend from a heuristic approach to mathematical programming-based approaches, as well as efforts to include more issues of the real complexity in snow plowing routing. They conclude that even when in the last decade there has been some impressive progress in snow plow routing research, similar progress has not occurred in implementing route optimization for winter road maintenance because it seems that models are generally still not comprehensive enough to consider all that needs to be included, and the mathematical programming-based models do not have the ease of use required by operating personnel. Eglese et al. (2014) provide detailed background on arc routing for the salt spreading. The survey shows that while early work has used simple constructive heuristics, more recently various metaheuristics algorithms have been developed for salt spreading applications. The authors note that there are few works of exact solution methods being used in real cases because a reasonable amount of computing time is required. Hence solution methods tend to rely on heuristic approaches. Two works are highlighted: Letchford and Eglese (1998) and Tagmouti et al. (2007), which are particularly relevant to the types of model and constraints found in spreading applications.

23 7 Road network monitoring and road marking are other activities of road network maintenance; however in arc routing problems they have not received as much attention as snow plowing or spreading chemicals for winter. The road network monitoring aims at maintaining the safety and the visibility of the roads and highways by a timely detection of the various incidents occurring on it. This problem was firstly modeled by Marzolf et al. (2006) as a periodic capacitated arc routing problem for the case where vehicles must inspect all the categorized road segments of a network over a two-week horizon. During a shift, a vehicle may have to leave the planned route to answer an emergency call and it may not be able to complete its planned itinerary. The authors rebuilt the routes using a mathematical linear formulation which selects pre-determined routes in order to maximize the number of passages on class 1 arcs. Later, Monroy et al. (2013) studied the same problem, but this time the objective is to build the initial plan minimizing the traveled cost and fulfilling frequencies of services on the roads classified into three categories. They present a mixed-integer program formulation and for larger instances they propose a heuristic solution method that works in two phases; first an assignation of arcs to shift is done, and then a rural postman problem is solved for each shift. In the Province of Quebec, road markings have to be painted or repainted every year. The Ministry of Transport uses a fleet of special vehicles to mark the roads and also tank trucks to meet the marking vehicles and replenish them. Amaya et al. (2007) introduce the capacitated arc routing problem with refill point, the problem consists in simultaneously determining the routes of marking vehicles and refilling vehicles that minimize the total cost. An integer linear programming model is presented and a cutting plane method to solve it. Later, Amaya et al. (2010) present a route-first cluster-second heuristic procedure for an extension of the problem considering multiple loads; in this version the refilling vehicle does not have to return to the depot each time it meets the marking vehicles. The time-sensitivity of the operations and the dynamic nature of the context are two important features on road winter maintenance noted in previous surveys. Eglese et al. (2014) and Perrier et al. (2012) affirm that the timing of operations is crucial to achieve the desired level of service in road winter maintenance. On the other hand, a real-time routing in road network maintenance is necessary to respond dynamically not just to atmospheric conditions and forecasts, but also to

24 8 equipment breakdowns, traffic congestion and accidents, all of which are more common in winter driving (Perrier et al., 2012). The survey from synthesis report on winter highways operations (Transportation Research Board, 2005) that included 22 prominent winter road maintenance agencies in North America highlights the use of dynamic routes in practice as 72% of the agencies indicated that they dynamically change routes. Table 2.1 summarizes the works on time-sensitive arc routing problems and dynamic arc routing problems in road network maintenance presented in the surveys. Table 2.1: Synthesis of works in road network maintenance considering time-sensitive and dynamic context Problem Authors Real application CARPTW* Eglese (1994) Spreading operations RPP** with deadline classes CARPTW* CARPTW* Rescheduling for Periodic CARP* CARP*** with dynamic information CARP*** with time dependent service cost CARP*** with time dependent service cost Letchford and Eglese (1998) Golbaharan (2001) Razmara (2004) Marzolf et al. (2006) Handa et al. (2005) Tagmouti et al. (2007) Tagmouti et al. (2010) Spreading operations Plowing routing Plowing routing Road network monitoring Spreading operations Spreading operations Spreading operations Problem characteristics Multi-depots Wide time windows Deadline classes Multi-depots Time windows Time windows Frequencies of service Reschedule of routes Requirements and demands change during the operation Time-dependent service cost Time-dependent service cost Solution method Two-phase heuristic Cutting-plane approach Column generation Column generation Mathematical formulations solved using CPLEX 8.0 Memetic algorithm Column generation Variable neighborhood descent Dynamic CARP*** with time dependent service cost Tagmouti et al.(2011) Spreading operations CARPTW*: Capacitated arc routing problem with time windows RPP**: Rural postman problem CARP***: Capacitated arc routing problem Dynamic version Time-dependent service cost Variable neighborhood descent

25 9 From Table 2.1, it can be seen that most works that consider timing-sensitive arc routing problems have focused on the capacitated case and column generation is the approach most used as the solution method. There are only three works that deal with dynamic aspects, in these cases heuristics and metaheuristics are proposed to solve the problem. Next sections correspond to contribution on time-sensitive (time windows) and dynamic arc routing problems where details of most of the works in Table 2.1 are presented. 2.1 Arc routing problems with time windows Arc routing problems focus on solving routing when the demands for services are located on edges or arcs of a network. Arc routing is the counterpart of vehicle routing, and addresses cases like waste collection where demands correspond to quantities to be collected in the streets, or salt spreading for ice clearance in winter where deliveries must be done over roads. When each customer e specifies a period of time, called a time window in which the service must occur, the conventional arc routing problems become arc routing problems with time windows. A time interval [a e, b e ] is specified for each customer, where a e 0 indicates the earliest arrival time and b e > 0 indicates the latest arrival time. Most of the arc routing problems with time windows consider the case when the service must start between the earliest and latest arrival time of the time windows. However there are other types of time windows: i) the service must finish no later than the latest time; in this case every time window begins at time zero and only deadlines are given to customers. ii) The service must be carried out during the interval of the time window. This is typical in cases such as trash collection where the collection must respect some schedules in large cities, or such as street sweeping where the activity must be done during some specific time because of parking restrictions. The Figure 2.1 presents the different types of windows. Other classification of routing with time windows considers soft and hard time windows. In the soft case, the vehicles are allowed to violate time windows but a penalty cost is incurred. On the other hand, in hard time windows a feasible solution must satisfy the time windows constraints for all the services and vehicles may wait at a node for service.

26 10 a e b e a. The service must finish before the latest time. a e b e b. The service must be executed during the time interval a e b e c. The service must start between the earliest and the latest time Figure 2.1: Types of time windows Time dependent arc routing problems can be seem as arc routing problems with soft time windows when the service cost is minimal within a given time interval and then increases linearly on both sides of the interval. The following sections present a brief description of the arc routing problems and a summary of the research that has been done considering timing-sensitive features The Chinese postman problem This problem was first suggested by the Chinese mathematician Kwan (1962). Formally, given a connected graph G = (V, E) where V is a set of vertices and E is a set of undirected edges, with distances on the edges, the problem is to find a tour, which passes through every edge in E at least once, starting and finishing at the same vertex, and in the shortest possible way. When the underlying graph is completely directed or completely undirected, the Chinese postman problem (CPP) can be solved in polynomial time (Christofides, 1973; Edmonds and Johnson, 1973). However when the underlying graph is mixed, the problem becomes NP-hard (Papadimitriou, 1976) The Chinese postman problem with time windows This problem is an extension of the CPP, where one vertex of the graph G is designed as the depot vertex, and the tour must start and finish at that vertex. In addition, time intervals are

27 11 introduced so that earliest and latest time constraints are specified for the start of service of each edge. The inclusion of time windows constraints makes the problem NP-hard in all the cases (Dror, 2000). The CPP with time windows (CPPTW) is studied at first by Wang and Wen (2002). They consider a mixed linear programming model of a directed CPP and incorporate the timeconstraints into the model. The formulation traces how to travel the network structure explicitly. The problem is defined on a directed graph G = (V, A) where V = {v i i = 1,, n} is a vertex set and A = v i, v j v i, v j v j, v i, v i, v j V is an arc set. Let T i k be the time that the postman visits vertex v i at k th iteration, assuming that the postman starts and finishes the tour at vertex 1, then T 1 1 means the postman s starting time at vertex 1, and T 1 K+1, K 1 is the completion time of the tour when K is large enough. D ii is the distance (time) from vertex v i to vertex v j and it is assumed that the triangle inequality holds for the distance measure. Let x ii k be a binary variable equal to 1 if at k th iteration the postman traverses from vertex v i to vertex v j, and k equal to 0 otherwise. The value of K must be given under the condition of K max (i,j) k x ii. Assuming that when a postman must traverse an arc more than once, the postman will deliver the mail when he traverses the arc at the first time, the model can be formulated as follows: minimize T 1 k+1 T 1 1 (2.1) s.t.: T k i T k k 1 D 1i x 1i (1, v i ) A, k = 1,, K (2.2) T 1 k+1 T i k D i1 x i1 k (v i, 1) A, k = 1,, K (2.3) T j k T i k D ii x ii k v i, v j A, i, j 1, k = 1,, K (2.4) k k (v i,v j ) A x ii = (v j,v i ) A x jj i = 1,.., n, k = 1,, K (2.5) x ii k k 1 v i, v j A, k = 1,, K (2.6) a i T i 1 b i i = 2,.., n (2.7) x ii k {0,1} v i, v j A, k = 1,, K (2.8)

28 12 The objective function looks for minimizing the total time for the postman to finish the tour. Constraints (2.2), (2.3), and (2.4) ensure the continuity of time. In (2.5) the condition that vertices must be symmetric is ensured. Constraints in (2.6) ensure that each arc must be passed at least once. Constraints (2.7) impose that postman has to arrive vertex i between time interval [a i, b i ] to deliver the mails. The variables are restricted to be 0-1 integer in (2.8). Note that this model does not consider waiting time once the postman starts the mail delivering. However, the authors modified the model a little to ensure the existence of the solution in the cases where waiting time is necessary and they employ the concept of fuzzy set theory when time constraints are not certain. This model includes a strong assumption: the iteration variable is a simple circuit. The problem is formulated as a circuit sequence such that each vertex in an iteration associates with only one starting time and in every circuit the depot vertex is included. Although the authors found an optimal value of K based on the even-degree properties of a directed CPP, this result is only valid for the case where the time windows can be fixed. They determined the bounds of the time intervals to get feasible solutions. However, if an arbitrary time interval is given, there can be no solution to this model. Aminu and Eglese (2006) studied the undirected case and modelled the problem using constraint programming. Two different formulations are proposed. The first formulation approaches the problem directly and the second transforms the problem to an equivalent vehicle routing problem with time windows (VRPTW). In these formulations it has been assumed that the time to travel over an edge is equal to the cost of travelling over the edge. First formulation (F1): Initially an arbitrary edge is added to represent the depot node, therefore the total number of edges in the graph is N + 1, if N is the number of the original edges in the graph. Three sets of decision variables are defined: i) EEEE = {E 1,, E N, E 0 } is the set of variables where E i indicates where e i E comes in the ordering for service and E 0 represents the order for servicing the edge depot (the domain of the EEEE variables is {1,, N + 1}). ii) a set of binary variables to indicate in which direction the edge is served; for each edge e i E, z i determines the direction that the edge is serviced (for e i = v p, v q, z i = 0 if the edge is serviced in the direction v p v q

29 13 and z i = 1 if the edge is serviced in the direction v q v p ). iii) a set of variables ccct i for e i E represent the cost (or time) to finish serving e i. Let ccct ii be the cost of the shortest path from the node where e i E was served last to the node where service starts for e j E. Let aaaat j be the actual given cost to service the length of e j E, and let ccct j be the cost to finish serving e j E. Then, for edge e j succeeding e i in the solution, the relationship ccct i + ccct ii + aaaat j ccct j holds. In addition, let a i and b i be the earliest and latest time to serve edge e i, and let u_cccc be the upper bound for the cost of the complete tour. F1 for the CPPTW is stated as follows: minimize ccct N+1 (2.9) s.t.: ccct i + ccct ii + aaaat j coot j i j, e i, e j E, when e j follows e i (2.10) a i ccct i b i fff i = 1,, N (2.11) 0 ccct N u_ccct (2.12) EEEE [1,, N + 1] (2.13) aaaaaaaaaaaa (EEEE) (2.14) E 0 = N + 1 (2.15) The objective function minimizes the time to complete service on all the edges of the graph starting and finishing at v 0. The cost of the current partial path is computed in (2.9), time windows constraints are defined in (2.10) - (2.12). The domain of the EEEE variables is given in (2.13) and (2.14) imposes that these variables must be all different. In (2.15), the variable representing the depot node is constrained to be served the last. The second formulation (F2) The second formulation is based on a graph transformation inspired by Pearn et al. (1987) where each edge of the graph is replaced with three nodes called the side and middle nodes. Using this approach the problem is transformed into a vehicle routing problem (VRP). The use of the middle node ensures that the three nodes representing an edge are serviced consecutively. However, the authors dispense the middle nodes and use just the side nodes to represent each

30 14 edge, and adding additional constraints. Demands of each edge side node are half of the demand on the edge they represent. The transformed problem may now be formulated as a special case of a clustered travelling salesman problem with time windows where each member of the cluster must be visited within its time windows before visiting a node in a different cluster. The clusters are formed by the side node pairs and additional constraints are added to ensure that the side nodes are consecutive node in any route. F1 found optimal solutions for problems with up to 15 edges and tight time windows. F2 was tested on a set of problems with up to 69 edges. Optimal solutions were found quickly when the time windows are tight. The results also show that as the time windows are made wider and the number or feasible solutions increases, some problems are not solved to optimality within a reasonable computing time The Chinese postman problem with time-dependent travel time In this problem the travel time on an arc depends on the starting time to travel it. Sun et al. (2013) studied the test sequence optimization in hybrid automaton (Springintveld et al. 2001), where the delay time of transition from state s i to s j is a function D ii (t i ) of the arrival time t i at s i. The authors model the problem on a directed network G = (V, A) with D ii (t i ) as the time dependent travel time of arc (v i, v j ) A. As each state s i corresponds to the vertex v i in V, and each transition from s i to s j corresponds to the arc (v i, v j ) in A, the optimal test sequence checking all transitions on the hybrid system is equivalent to a minimum time dependent CPPtour that traverses all the arcs in time dependent network G. The problem is formulated as an integer programming model which uses the iteration variable introduced by Wang and Wen (2002) to trace the tour. To solve the problem the authors propose a cutting plane heuristic algorithm. They present cutting planes that can be separated polynomially using a maximum flow algorithm. In addition two upper bound heuristics are also designed to terminate the cutting plane procedure whenever the current LP solution is fractional and violates no inequality. The algorithm is tested on a set of generated instances with up to 25 vertices and 50 arcs. The computational results show that the lower bound obtained by adding cutting planes improves the LP relaxation bound of the original formulation for all instances. The gap between the lower bound and the best upper bound for all the tests ranges between 0.90% and 31.75%.

31 15 Later, Sun et al. (2015), propose an integer programming approach, an extension of the previous formulation. The new formulation does not assume that every cycle in the graph must visit the depot. They use two sets of constraints: the first part has a strong combinatorial structure, which is linear and refers to the routing; the second part is related to time-dependent travel time and is not linear. In the case when all the travel times are piecewise functions of the starting time, a linearization is provided. The formulation is solved with a cutting plane algorithm. The algorithm was tested on a real-world and several randomly generated instances. For the real instance with 27 vertices and 27 arcs the algorithm finds a gap between the lower and upper bounds is less than 11.80% in 425 s. For the set of generated instances with between 10 to 25 vertices and 20 to 60 arcs, and piece functions with 2 to 4 times intervals, and fluctuation interval [ R, R] from [ 10,10] to [ 30,30]. The results indicate that the face defining and valid inequalities proposed play an important role when the number of time intervals and the scale of fluctuation become larger Time dependent Chinese postman problem with time windows Sun et al. (2011) consider a variant of the CPPTW, where the travel time and service time on arcs depend on the starting time. The travel and service times are piecewise linear functions of time. The problem is defined on a directed graph G = (V, A), where V is the vertex set, and A is the arc set. Each arc (v i, v j ) A has an associated time window a ii, b ii, with a ii, b ii Z + {0}. The travel time D ii (t i ) Z + of an arc (v i, v j ) A depends on the starting time t i. Similarly, let S ii (t i ) Z + be the time dependent service time of arc (v i, v j ) A with starting time t i. The problem looks for finding the minimum total travel time tour starting at the given depot vertex v 0 V and starting time t 0 and passing through each (v i, v j ) A at least once, such that the completion time of servicing each arc in A is in its associated time window. The authors present a graph transformation method through which the time dependent CPPTW can be reformulated as a 0/1 integer linear programming without any timing constraint. The problem is transformed into a generalized rural postman problem. The algorithm is tested on five sets of randomly generated instances, where V ranges from 30 to 50, and A ranges from 50 to 130. They were able to solve to optimality instances with up to 100 arcs within 15 minutes plus

32 16 the time of transforming the graph. In addition, they tested four sets of instances to find how the width of time windows can affect the computational time The rural postman problem This problem was introduced by Orloff (1976). Formally, the rural postman problem (RPP) is defined on an undirected connected graph G = (V, E), where V is the vertex set and E is the edge set. A subset R of the edges are required. Each edge e has a cost c e 0. The RPP consists of finding a minimum cost tour in G traversing every edge in R at least once. The RPP historically arose in rural mail delivery; however applications of the RPP take place in contexts where some edges of a graph must be serviced by an uncapacitated vehicle. By reducing the NP-hard Hamiltonian cycle problem to the RPP, Lenstra and Kan (1976) showed RPP to be NP-hard, except when R = E, in which case the RPP becomes the CPP The RPP with deadline classes The RPP with deadline classes is presented by Letchford and Eglese (1998). In this problem the set of arcs are partitioned into small number of classes according to priority, with each class having its own deadline on service. Formally the set R of required edges is partitioned into {R 1,, R p } and services for each class k of edges (k = 1,, p) must be completed by time T k. The problem is formulated on an undirected graph as an integer linear programming model. The authors have proposed several classes of valid inequalities which exploit the structure of the problem. They used the dual cutting-plane method (Nemhauser and Wolsey, 1988) to solve the problem: an initial LP relaxation is solved and then each time that a violated inequality is identified, it is added to the LP, and the LP is solved using the dual simplex method. When no more violated inequalities can be found, branch-and-bound is invoked to obtain integrality. The algorithm was tested on a set of 10 instances: 5 problems from Corberán and Sanchis (1994) were adapted and the value of p was set to 1 and 2 for each of the problems. The problems have between 22 and 67 required edges and between 3 and 6 connected components. The cutting plane algorithm showed a good performance as all the instances were solved to optimality.

33 The time-dependent rural postman problem Applications involving scheduling with time-dependent processing time (Alidaee and Womer, 1999; Sundararaghavan and Kunnathur, 1994) motive this problem. The travel (or service) time of each arc depends on time, that is, the travel time of an arc depend on the time interval during which the arc is traversed, and the postman is not required to cover every arc in the network, only a subset of arcs. A formal definition is presented by Tan et al. (2013). Let G = (V, A) be a directed graph, where V is the vertex set and A is the arc set, which includes a subset of required arcs A R that must be serviced. Each required arc in A R is associated with a travel time and a service time, while the arcs not in A R have travel time only. Both the travel time and the service time are time-dependent piecewise functions. Let tt ii (t i ) and st ii (t i ) denote the time-dependent travel time function and the service time function, where t i is the time at the beginning of travel along an arc or the service time on an arc (v i, v j ). A postman is required to service the arcs in A R and is located at the depot vertex from which to start and end the service tour. The postman is allowed to wait along the tour and must start after a given time t 0. The time-dependent RPP consists of finding a tour servicing all of the required arcs with a minimum cost with respect to the time-dependent travel time and the service time. Tan and Sun (2011) and Tan et al. (2013) present the version that considers only timedependent travel times and propose an arc-path formulation and strong valid inequalities. The authors note that the constant travel time assumption in other timing sensitive arc routing problems never holds on the time dependent network. Thus the transformation methods which use the shortest path algorithm have as sub-problem the time dependent shortest path problem, which has been proved to be NP-hard (Orda and Rom, 1990). They propose an integer linear programming: an arc-path formulation for the problem with a constraint set divided into two parts. The first part defines the polytope of the arc-path alternation sequence and the second part is closely related to time-dependent travel time. The service time functions are considered as piecewise functions, and are linearized. Based on the polyhedral results, a cutting plane algorithm was proposed as solution method. The algorithm was tested on two sets of randomly generated instances with up to 50 vertices and with up to 50 arcs; the travel time is treated as the step function with 3 and 4 intervals, and the percentage of required arcs ranges from 10% to 30%. The computational results show that for all 42 test instances the method solved instances up to 25

34 18 vertices and 50 arcs. The relative gap between the best feasible solution and the lower bound is 3.16%for all the instances on average Capacitated arc routing problem This problem was introduced for the directed case by Golden and Wong (1981), and later (Belenguer and Benavent, 1991) formulated the problem for an undirected graph. It generalizes the Chinese postman and rural postman. Given an undirected graph G = (V, E) with V as the set of vertices and E as the set of edges. A subset R of edges are required. In addition, let c e 0 be the edge cost, d e 0 be the edge demand for every e E, and Q be the vehicle capacity. The capacitated arc routing problem (CARP) consists of determining a minimum cost traversal of all edges in R, so that each vehicle starts and finishes at the depot vertex v 0 V, and the total demand of all edges serviced by any particular vehicle does not exceed its capacity Q. The CARP defined on two specific graphs is not NP-hard, but on the other cases is NP-hard (Busch, 1991) The CARP with time windows The CARP with time windows (CARPTW) is defined as the classical CARP with the extra requirement that the service of each demand edge must begin within some pre-specific time window. Given an undirected connected graph G = (V, E), where c e 0 is the cost of traversing the edge e = (i, j), and d e 0 is the demand of the edge. A number of vehicles, each of capacity Q are placed at the node depot v 0. Let t e denote the time needed to traverse edge e, and let each demand edge have a time windows [a e, b e ], in which service of the edge must start. The problem consists of finding tours such that i) All edges with d e > 0 are serviced, ii) Vehicle capacities are respected, iii) Service of each demand edge start within the time windows of that edge, and iv) Total cost is minimized. Eglese (1994) firstly presents an application in routing for winter gritting: local authorities treat the roads by spreading a de-icing agent on them, multiple depot locations and limited vehicle capacities are considered, and roads with different priorities implies that some roads must be treated within two hours and other within four hours of the start of gritting. This problem can be considered as CARPTW, where the time windows are rather wide. A two-phase heuristic method is proposed as solution method: first the optimal solution of an unconstrained CPP for

35 19 the network considering just category 1 roads is found, depot locations are specified and routes are defined, second a simulated annealing algorithm attempts to improve the current solution. Mullaseril (1997) studies the problem of managing a fleet of trucks for distributing feed in a large livestock ranch in Arizona. The ranch produces cattle. The cattle are kept over a large area in approximately 500 pens with rectangular shape. The pens are arranged in rows by a network of paved and dirt roads. The feed type, volume and feeding time for each pen may vary from day to day because there is a constant movement of cattle in and out of the yard. A route may be stipulated to deliver the exact demand to several pens, but weighing inaccuracies may force it to only partially supply the last pen and that pen would need to have feed delivery from more than one route. Thus, the feed-yard allows split-delivery. In the problem case there is a further consideration. A vehicle may traverse the arcs at two different speeds-discharging speed and dead heading speed. The author models the feed delivery problem for the cattle ranch as a collection of capacitated rural postman problem with time windows and split delivery. The livestock ranch is represented as a connected mixed graph, where the set of requirements are arcs (because the design of the delivery trucks). The author presents heuristic algorithms for obtaining fast solutions for this class of problems. Also he presents solution strategies for obtaining tight lower bounds to the optimal solution and optimal solutions to some of the real life split delivery problems with time windows. In addition, the author presents a transformation of this arc routing setting into an equivalent node routing problem where the number of nodes is the same as the number of required arcs in the original arc routing problem. The problem in the equivalent graph of nodes is decomposed and solved with a column generation approach. The master problem is formulated as a set covering problem based on the work of Desrosiers et al. (1995). The sub-problem is modeled as a shortest path problem with resource constraints and solved using dynamic programming algorithm as an extension of the work of Desrochers (1988). This approach shows to be successful on instances up to 55 nodes. Gueguen (1999) describes an integer linear programming model for the undirected CARPTW and another transformation into a VRPTW, but without numerical results. The author presents the only direct model for an undirected CARPTW, which is presented as follows:

36 20 It is known that an optimal solution to the undirected CARP always traverses each edge at most once per direction in each route. The example of Figure 2.2 shows that this property is not true when time windows are considered. In the example, all costs and traverse times are equal to one, there is not service time considered, and the capacity is not restricted for a vehicle that must start and finish at the depot (black node) and visit all the edges in their time intervals. From the example, the only solution to the problem with a single vehicle is: starts at the depot, serves in this order the edges 5, 1, 4, 2, 3 and finishes at the depot. The vehicle traverses edge 3 six times. [11,12] [8,9] 2 [13,14] [5,6] [2,3] Figure 2.2: Example of time windows in an undirected graph (Source: Gueguen (1999)) Based on the previous analysis, Gueguen (1999) proposed a model making (m + 1) copies of each edge, if the number of edges of the graph is equal to m. This number is an upper bound considering that in the worst case an edge e will be traversed: Once for serving each of the other (m 1) edges Once for serving the edge e Once for coming back to the depot. Having a maximum of (m + 1) times. Three types of decision variables are defined: traverse, service, and time variables k x iiii =1 if vehicle k traverse the copy β of edge j immediately after copy α of edge i, and 0 otherwise. These variables are only created when the final node from edge i is the same as the initial node from edge j. s i k =1 if vehicle k services the edge i, and 0 otherwise. t ii k is the time to start traversing copy α of edge i by vehicle k.

37 21 The CARPTW formulation is as follows: m i=1 m j=1 m+1 β=1 k k x iiii minimize K m+1 k=1 =1 c pi + K m k k=1 c si s i (2.16) s.t.: m m+1 k m m+1 k j=1 α=1 x jjjj = j=1 α=1 x iiii i = 1,, m β = 1,, m + 1 k = 1,, K (2.17) K k=1 = 1 i = 1,, m (2.18) s i k m k i=1 q i s i Q k = 1,, K (2.19) m m+1 k s k i j=1 α=1 x i1jj i = 1,, m k = 1,, K (2.20) i=1 T k k k ii + t pi C 1 x iiii T jj i = 1,, m j = 1,, m β = 1,, m + 1 α = 2,, m + 1, k = 1,, K (2.21) T k k i1 + t pi + t si C 1 x i1jj T k jj i = 1,, m j = 1,, m β = 1,, m + 1 k = 1,, K (2.22) T i1 k a i i = 1,, m k = 1,, K (2.23) T i1 k b i i = 1,, m k = 1,, K (2.24) k x iααα {0,1} i, j = 1,, m α, β = 1,, m + 1 k = 1, K (2.25) s i k {0,1} i = 1,, m k = 1,, K (2.26) t ii R + i A, k = 1, m (2.27) The objective is to minimize the traverse and service cost. It is assumed that if a service is done in an edge, it is done in the copy 1 of that edge. The constraints (2.17) are conservation flow, the constraints (2.18) and (2.19) make sure that the each edge is serviced, and the vehicle capacity is not exceeded. Constraints (2.20) make a vehicle traverses the copy 1 of an edge if the vehicle serve that edge. Constraints (2.21)-(2.24) set the accumulated times for all copies of the edges and satisfy the time windows when edges are serviced. C is a large constant with value greater than the longest time route. The three sets of variables are defined in (2.25)-(2.27).

38 22 As the author noted, the model is not practical, then he proposed a transformation to node routing problem based on the transformation presented by Mullaseril and Dror (1996). Golbaharan (2001) studies a multi-depot CARPTW in the context of snow removal. Every snow plow starts from a depot and returns to the same depot. The problem is formulated as a linear integer programming model; indeed as a constrained set covering problem. The objective function in this case minimizes the total cost of the routes and the penalty for using extra snow plows. A column generation method is implemented to solve the problem. The master problem includes the constraints on the number of snow plows available at each depot and the constraints which guarantee that a required road segment is serviced. The sub-problem contains the time window constraints and network flow constraints. The master problem is solved by the dual simplex method, and the sub-problem for every depot is formulated as a shortest path problem with time windows associated with the edges in the network, and it is solved with a label-setting algorithm. Computational experiments were conducted on real-life instances involving 7 depots, 21 snow plows, 362 nodes, and 707 required edges. Razmara (2004) presents a real problem of the Swedish National Road Agency on snow removal routing for homogeneous snowplows; in this case every segment in the network must be plowed in its associated time windows and the routes must start from an end at the same depot. The case is formulated as a linear integer programming problem and solved using a Dantzig- Wolfe decomposition scheme. The master problem is formulated as a constrained set covering problem. The sub-problems are resource constrained shortest path problems, where time is the resource. The sub-problems are solved by a label-setting algorithm. An integer solution to the master problem is found by a greedy algorithm or a variable reduction procedure. Later, Wøhlk (2005) provides two mathematical models for the undirected CARPTW, one based on constructing a node duplicated network on which an integer linear programming is built and one based on a transformation into an equivalent node routing problem, the VRPTW. Several heuristics and a dynamic programming algorithm combined with simulated annealing, called DYPSA, are proposed. The average DYPSA performance is about 8% above lower bounds. The author also presents a column generation method to get tight lower bounds. Reghioui et al. (2007) suggest a greedy randomized adaptive search procedure (GRASP) with path relinking for the undirected CARPTW. Two constructive heuristics are used in the GRASP

39 23 algorithm: randomized path-scanning heuristic and randomized route first cluster second heuristic. Local search (OR-OPT, SWAP, and 2-OPT) is used to improve each solution. Path relinking is used as an intensification strategy working on a small pool of elite solutions collected during the GRASP. Computational results showed that the algorithm found 17 optimal solutions (including 4 new ones) on the set of 24 instances proposed by Wøhlk (2005) and the gap to lower bounds is less than 1%. Johnson and Wøhlk (2009) propose two column generation method and a heuristic approach. The master problem is a large set partitioning problem, and the sub-problem finds feasible routes with respect to both capacity and time windows. Two methods are presented to solve the set partitioning problem: a two-phase method and an iterative method. The heuristic approach generates only tours that are good in some pre-specified sense and subsequently use the presented methods for solving a set partitioning problem based on these columns. The method is tested on a set of 20 instances adapted from the so-called Eglese instances for the classical CARP. The instances contain from 51 to 190 required edges. The iterative method is superior to the twophase method both regarding computation time and ability to solve hard instances. The case where the time windows are soft and violating these implies some extra cost is studied by Afsar (2010). A Dantzing-Wolfe decomposition and column generation approach to solve the problem optimally is presented. The sub-problem is a non-elementary capacitated shortest path problem. Computational results are reported on a set of modified instances of Johnson and Wøhlk (2009). The problems have up to 40 nodes and 69 required edges and linear and symmetric penalty costs were considered Capacitated arc routing problem with time-dependent service cost Tagmouti et al. (2007) study the directed capacitated arc routing problem with timedependent service cost inspired on winter gritting operations, where the timing of an intervention is crucial; if the intervention is too early or too late, the cost in material and time increases. In this case the cost of service depends on the time of beginning of service, indeed, the cost is a piecewise linear function of time. The problem consists of finding a set of routes that serve all required arcs in the graph at least cost (sum of travel cost and service cost) with the constraint that vehicles are not allowed to wait along their route and must be back at the depot by a given deadline. A column generation algorithm is proposed to solve the problem where the master

40 24 problem is a set covering problem and the sub-problems are time-dependent shortest path problems with resource constraints. The method is tested on a set of instances derived from Solomon s problems of the VRPTW (Solomon, 1987). Instance with up to 40 required arcs were solved to optimality. Later, Tagmouti et al. (2010) propose a variable neighborhood descent (VND) heuristic for solving the problem. Two initial solutions are constructed with an insertion heuristic and an adaption of the savings heuristic (Clarke and Wright, 1964). The VND is then applied to each solution to improve them. The structure of the neighborhoods manipulates an arc or sequence of arcs. They tested the performance of this algorithm on a set of instances adapted from the CARP instances of Golden et al. (1983), Li (1992), and Li and Eglese (1996). The algorithm behaved appropriately and showed to be fast and competitive when compared with the adaptive multi-star local search algorithm of Ibaraki et al. (2005) which is designed for the VRP with soft time windows. 2.2 Dynamic arc routing problems Several real-life routing problems have been studied in a static context, where it is assumed that all data about the problem are known in advance. However a number of technological advances have made possible that the information available to the planner be updated during the execution of the routes. In this case, part or all of the input is unknown and revealed dynamically during the execution of the plan. Although there is a fair number of papers on dynamic node routing problems as it is presented in the survey of Pillac et al. (2013), dynamic arc routing problems have not received enough attention of the research community. This section summarises the works that deal with dynamic issues in arc routing problems Dynamic rural postman problem Moreira (2007) introduce a dynamic RPP motivated by an industrial application on a high precision tools factory where pieces of different shapes have to be cut out of a surface by means of an electrified string. In a first phase, the pieces are nested in the given surface. In a second phase, an optimal cutting path is desirable for cutting out of pieces. The cutting surface is

41 25 suspended, which involves the falling of the surface portions cut out in succession. Any movement of the cutting tool generates an effective cut, unless of course it is performed over a region already cut out. Therefore the graph in which the cutting path is determined changes in a dynamic fashion along the cutting process itself. There are two considerations: every piece that is not completely cut out must not be traversed in its interior and there may not remain uncut pieces in areas that have been cut out. The problem is modeled as a rural postman problem where the cutting cost is minimized. Two heuristics are proposed to solve the problem. The first, called higher up vertex chooses the higher up vertex among the candidates reachable by uncut edges. The second, called minimum empty path chooses the nearest candidate as next vertex among the visible candidates. The two heuristics were tested on 10 real industrial instances containing between 19 and 52 pieces, and between 183 and 639 vertices. The proposed heuristics showed improvements of about 3% with respect to actual solutions Dynamic CARP Dynamic issues in the CARP have been addressed in a few works. Handa et al. (2005) present an application in salt spreading on a road network in south Gloucestershire, England. The routing requirements on a particular day are linked to road weather information system (RWIS), which predicts road surfaces temperature and conditions across the road network within 20 minutes intervals. The information of road requiring treatment and the amount of salt needed for each road may change for each shift and even during a working shift. The problem is modeled as a classical CARP. A prototype system is proposed and consists of the RWIS and an evolutionary salting route optimization module. The optimization module has a memetic algorithm as optimization method. The algorithm uses the crossover EAX operator (Nagata, 1997) due to its search ability and a repair operator for offspring individuals is incorporated because the search space may include solutions that exceed the vehicle capacities. The prototype was examined on two typical shifts (one with 385 required edges and 11 vehicles and other with 97 required edges and 3 vehicles). A CARP is solved at the beginning of each shift with the predicted data. The results showed the effectiveness of the proposed method. Yazici et al. (2014) present an interesting application for the multi-robot sensor-based coverage problem. A path planning must be designed such as every point in a given workspace is covered at least once by one of the robot s sensor. Initially, the robots are assumed to be at the

42 26 same starting point with equal energy capacities, but due to partially unknown nature of the workspace, the robots may face blockage on routes, and a fast re-planning is necessary considering remaining capacities and current positions of the robots. The problem is modeled as a CARP and the new plan is obtained by a modified Ulusoy s partitioning algorithm (1985). The algorithm was tested on two different real environments: on an indoor laboratory with 20 vertices and 2 robots, and on a larger laboratory with 90 vertices and up to 10 robots. The authors determined the maximum number of robots to be assigned to a given coverage task with efficient coverage cost. Tagmouti et al. (2011) study the dynamic CARP with time-dependent service cost. This work is an extension of their previous works (Tagmouti et al., 2007, 2010) where the same problem is considered but on the static case. In the dynamic case the time interval where the service cost is minimal changes due to weather report updates, and therefore real-time modifications are required to the current routes. An adaptation of the VND of their previous work is presented as solution method. A starting solution is first computed with VND using service time cost functions based on an initial forecast. A simulated storm goes through the network and move along the x-axis. At different times, weather reports are received and update the storm speed. The VND is applied on a new static problem each time a weather report is received. In each static problem the graph is updated taking into account the already visited arcs. The algorithm was tested on a set of 60 generated instances with weather reports received every 5 minutes. The VND showed to be fast and allows the system to quickly use the new solution. Weise et al. (2012) present a developmental solution method to the CARP that is suitable for dynamic scenarios. The method is based on genetic programming; it works with a solution space defined by all possible tours represented as a permutation of a subset of the required edges but a search space different to the solution space. The approach iteratively adds edges to a solution based on an environment s state. The main objective is to minimize the total traverse cost without consider limit on the number of vehicles. The solution method was tested on a large set of CARP benchmark instances for the static case. Some scenarios were derived for the dynamic case from 2 instances with 25 and 66 required edges by removing a certain number of edges and solving the problem with the new set of requirements.

43 27 Recently Liu et al. (2014) presented a memetic algorithm for the dynamic CARP. The algorithm is capable to solve CARP with variations in vehicle availability, road accessibility, new added task, canceled task, variation of traffic conditions, and variation of demands. The memetic algorithm incorporates a new split scheme with a path repair operator in order to handle the attributes of the problem. It combines features from global and local search, and has four key steps: split method, parent selection, crossover, and local search. 4 dynamic problems taken from Min et al. (2014) were used to test the algorithm including a case of 10 nodes, and 3 cases of 100 nodes with up to 33 tasks initially. The working of the proposed algorithm is illustrated using the 10-node example. The algorithm showed great potential for solving realistic dynamic CARP problems.

44 28 CHAPTER 3 ARTICLE 1 : THE RURAL POSTMAN PROBLEM WITH TIME WINDOWS Marcela. Monroy-Licht 1,2, Ciro Alberto Amaya 3, André Langevin 1,2 1 Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, Canada 2 Centre de recherche sur les réseaux d entreprises, la logistique et le transport (CIRRELT), Montréal, Canada 3 Departamento de Ingeniería Industrial, Universidad de Los Andes, Bogotá, Colombia Abstract The rural postman problem with time windows for the undirected case is introduced. The problem occurs in the monitoring of roads for black-ice detection. Different formulations are proposed and tested on sets of instances adapted from the literature. A cutting plane algorithm based on valid inequalities for the traveling salesman problem (TSP) with time windows and the precedence constrained TSP is presented as a solution method and tested on a set of real-life networks. Computational results show that this approach is able to solve to optimality instances with up to 104 required edges. At the end of the article the formulations for the undirected case are extended to the directed case. roads Keywords: Rural postman problem, time windows, cutting plane algorithms, monitoring of 3.1 Introduction The rural postman problem with time windows (RPPTW) involves finding a minimum- cost tour that goes through a set of required edges in a network. A vehicle leaves the depot, visits the required edges, and returns to the depot. A release time and a due time are given for each required edge. The tour is feasible if the visits are carried out during the defined time windows. Waiting times are allowed, i.e., the vehicle may arrive at any required edge earlier than its release time, but the service cannot start until the time window opens. Costs of service are associated

45 29 with required edges and traversal costs with non-required edges. The vehicle may go through a required edge more than once, and the cost is normally lower when it does not service the edge. The real-world application underlying this study is the monitoring of roads for black-ice detection (black-ice is a thin coating of clear or transparent ice on the pavement which is difficult for the drivers to see). This activity is carried out by the Ministry of Transport in the province of Quebec from mid-october to mid-december. The goal is to check the state of roads and take measures to prevent accidents. During this period, black ice on roads is almost invisible to the users; timely detection avoids pedestrian falls or automobile accidents. Currently, a patrol must cover a network and generate reports about the state of the roads. The available information refers to a short-term weather forecast and a characterization of roads with high likelihood of black-ice formation; for example it is known that bridges and roads located near rivers are particularly susceptible to ice formation under certain meteorological conditions. The road segments to be checked are located in areas where the weather forecast indicates low temperatures and rain. The weather forecast induces the time windows for monitoring some of the road segments over a large region. Normally the patrol has enough time to visit all the roads defined previously in the schedule. The RPPTW reduces to the rural postman problem (RPP) when the time-window constraints are not taken into account, so it is NP-hard. Little attention has been paid to the RPPTW. To the best of our knowledge the works of Nobert and Picard (1994) and Kang and Han (1998) are the only two related to the non-capacitated case and the one of Mullaseril et al. (1997) presents the capacitated version. Nobert and Picard (1994) introduce a heuristic algorithm for the RPPTW. In their problem the required arcs are of two types: arcs that must be visited during the morning and arcs that may be visited all day long. They propose a heuristic method based on the solution of two rural path problems and on the computation of appropriate penalties. Numerical results are not published. Kang and Han (1998) consider the problem as a multiobjective optimization problem because they allow arrival at the required arcs after the due times, which incurs a cost penalty. The objective is to reduce the total traveling cost and total penalty. The authors present a genetic algorithm and compare three crossover operators.

46 30 Mullaseril et al. (1997) describe a feed distribution problem encountered on a cattle ranch in Arizona. The problem is cast as a collection of capacitated rural postman problems with splitdeliveries and time windows. They present some heuristics and compare them with the working practices on the cattle ranch. Another related problem, the RPP with deadline classes, has been studied by Letchford and Eglese (1998). They consider a single-vehicle arc routing problem in which the required edges are partitioned into a number of classes according to priorities, each class having its own deadline. An optimization algorithm is presented based on the use of valid inequalities as cutting planes. They tested the algorithm on a set of instances for the RPP from Corberán and Sanchis (1994) and found optimal solutions for all cases up to 67 required edges. Our work addresses the undirected version of the problem. We assume that the costs of service are equal to the traversal costs, but our approaches could be easily modified if this is not the case. Our main contribution is to present the problem for a real-life application and several ways to model it. Three formulations are presented: one where the decision variables explicitly express the number of times an edge is traversed and two based on graphs equivalent to the original one. We then explore the third formulation, obtained when we transform the original problem to a traveling salesman problem with time windows and side constraints. For the traveling salesman problem (TSP), polyhedral approaches have been extremely successful (Fischetti and Toth, 1997; Jünger, et al., 1995; Padberg and Rinaldi, 1991). Ascheuer et al. (2001) solve the asymmetric TSP with time windows (ATSP-TW) by a branch-and-cut method; they solve in a satisfactory way real-world instances of the control of a stacker crane in a warehouse. We have chosen to use the polyhedral approach. The RPPTW is formulated as an integer linear program that is solved by a cutting plane algorithm. The paper is organized as follows. Section 3.2 presents three different models for the undirected case. In Section 3.3 we summarize the valid inequalities that we use as cutting planes in our algorithm. We briefly describe the solution algorithm in Section 3.4. Section 3.5 outlines the computational experiments. In Section 3.6 an extension of the formulations is given for the directed version of the problem, and Section 3.7 provides concluding remarks.

47 Undirected RPPTW Let G(V, E) be an undirected graph, where V is the set of vertices and E is the set of edges. Given a subset E R E of required edges to service, the problem of finding a minimum cost tour traversing at least once all the required edges is known as the RPP. In the RPP each edge e E R is serviced exactly once, but can be traversed an additional number of times in a deadheading mode. Christofides et al. (1981) and Corberán and Sanchis (1991) present two formulations that use decision variables x e = number of times edge e is replicated in the optimal RPP solution if e E R, and x e = number of times edge e is traversed if e E E R. Although x e can be bounded above by 1 if e E R, and by 2 if e E E R (Eiselt, Gendreau, and Laporte, 1995), when time windows are considered this results is not valid. Indeed an edge could be traversed E R + 1 times in the worst case (Gueguen, 1999). An extension of these RPP formulations to RPPTW is difficult because it is not possible to associate a unique starting and completion time for an edge e E R. Arc routing problems with time windows are very hard to model directly without an extensive graph modification (Dror et al., 1997; Mullaseril, 1997; Mullaseril and Dror, 1996). We turn on alternative modeling approaches. Some arc routing problems formulations use binary decision variables connecting edges (x ii = 1 if edge j is traversed after edge i) or nodes (x ii = 1 if node j is traversed after node i). We propose three formulations for the problem. The first considers a formulation based on edge linking variables. The other two formulations are based on transformed graphs and the decision variables linking nodes. The first transformation considers the required edges as nodes and joins them by means of arcs that represent the shortest paths among them in the original graph. The second transformation considers the nodes incident to the required edges and connects them with an arc that again corresponds to their shortest paths in the original graph Model on the edges This formulation is based on the work presented by Gueguen (1999). He proposes a mixed integer program (MIP) formulation for the capacitated arc routing problem with time windows (CARPTW). Apparently this is the only formulation on edges for the CARPTW; however, the

48 32 author does not present numerical results. We modify Gueguen s formulation by adding a duplicate of each required edge to keep track of the direction in which a vehicle must travel along these edges. This is necessary to guarantee conservation of flow on the nodes of the network. Consider the graph G and the subset E R E previously defined. Let A be the set of edges that includes E, a duplicate i of each required edge i E R, and an artificial edge "e 0 " that represents the depot. The duplicate edges have the same cost and time windows as the originals. Let P be the set of pairs of edges {i, i } such that i, i correspond to the same required edge (the order i, i is determined arbitrarily), and R the set that contains all i E R and their duplicates. Additionally, c i is the traversal cost of edge i, T i is the traversal time of edge i, [a i b i] is the time window for edge i; and for the edge "e 0 " we set c e 0 = 0, T e 0 = 0, and a e 0 = 0. M is a large integer number, which could be bound by M = max i E T i E E R + 1, and δ i is the set of vertices incident to edge i. m = E R + 1 is the maximum number of times an edge can be traversed. The decision variables defined hereafter allow us to keep track of the number of times (each time corresponds to a copy of an edge) that the vehicle traverses each edge. Let the decision variables x iiii =1 if copy l of edge j is traversed immediately after copy k of edge i, and 0 otherwise; and let t ii be the time to start traversing copy k of edge i. The formulation on the edges is as follows: m m minimize c i x iiii i A j A k=1 l=1 δ j δ i,j i (3.1) s.t.: j A δ j δ i,j i m x ii1l + x i j1l l=1 j A δ j δ i,j i m l=1 = 1 {i, i } P (3.2) j A δ j δ e 0,j e 0 m x e0 j1l 1 (3.3) l=1

49 33 m x iiii i A k=1 δ i δ j,i j m = x jjjj i A k=1 δ i δ j,i j j A, l = 1,, m (3.4) m m x iiii + m m x i jjj 1 {i, i } P (3.5) j A k=1 l=1 δ j δ i,i j,j i j A k=1 l=1 δ j δ i j,i,j i t ii + T i t jj + M 1 x iiii i A, j A δ i δ j, i j, j e 0 k, l = 1,, m (3.6) t i1 a i i R {e 0 } (3.7) t i1 b i i R (3.8) m x iiii l=1 1 i A, j A δ i ( ) = δ j (+), i j, k = 1, m (3.9) x iiii {0,1} i A, j A δ i δ j, i j, k, l = 1,, m (3.10) t ii R + i A, k = 1, m (3.11) The objective is to minimize the total traversal cost. Services are ensured by constraints (3.2). Constraint (3.3) forces the tour to start at the depot. Flow conservation is ensured by constraints (3.4). Constraints (3.5) avoid solutions with subcycles between any pair of edges in P that represents the same edge in A. Inequalities (3.6), (3.7), and (3.8) are the time-window constraints. Constraints (3.9), not in Gueguen (1999), allow us to reduce the number of equivalent solutions. Finally, constraints (3.10) and (3.11) define the decision variables. In this model, the original graph is extended firstly by making a duplicate of all required edges, secondly by making E R + 1 copies of all edges. If the number of required edges is large, the formulation is intractable. The model also uses a large real value M that generates weak relaxations and numerical difficulties in the solution methods.

50 Model on the required edges Since the model on the edges (Section 3.2.1) is not practical, we propose an equivalent formulation. On the graph G = (V, E) two required edges can be connected successively on a route in four ways, depending on the traversal direction of the two edges. This leads us to define the problem on a new graph G 0 = (N, A 0 ). For each required edge i in G we define two nodes i, i in G 0, where i, i N represent the two possible directions in which edge i in G can be traversed. Each arc of A 0 connects a node i N with a node j N if they do not correspond to the same edge in G. The cost of the arc that connects node i to node j in G 0 is equal to the cost of the edge represented by i plus the length of the shortest path in G from the final node of the edge represented by i to the initial node of the edge represented by j, according to the traversal directions. Figure 3.1 shows an example of the transformation. The original graph is presented in a). The depot is located at the black node; the edge indices are shown near each edge; and the traversal costs are in parentheses. There are three required edges (2, 4, and 5). The transformed graph is illustrated in b). Its nodes are labeled with the same number as their corresponding required edges. Arrows over and under the nodes indicate the traversal direction that each node represents. Note that pairs of nodes corresponding to the same required edge are not connected (10) 2 (2) (1) (4) 5 (1) 4 (2) (3) 5 4 Required edges a) Original graph b) Transformed graph Figure 3.1: Transformed graph for the model on the required edges

51 35 Finally we add an artificial node labeled "0" to represent the depot. We join "0" to all nodes in N by means of two arcs with a cost equal to the length of the shortest path in G from the depot to the edges, and from the edges to the depot respectively, again according to the traversal direction. Table 3.1 indicates the cost of the arcs of the transformed graph. The time window for each node is the same as for the corresponding edge. Table 3.1: Costs of the transformed graph Model on the required edges To From In the transformed graph G 0 we look for a minimum-cost tour that visits one of the two nodes that represent the same required edge. The tour starts and ends at the depot, and the visits must satisfy the time windows. Let us consider the set C that includes the pairs of nodes {i, i }, where i, i N, such that i, i represent the same required edge. We consider the following parameters: c ii is the traversal cost from node i to node j, T ii is the traversal time from node i to node j, [a i b i ] is the time window for node i, and M ii = max b i + T ii a j,0. We set a 0 = 0 for the depot node. We define the decision variables x ii to be equal to 1 if node j is serviced immediately after node i, and 0 otherwise, and t i to be the arrival time at node i. The formulation on the required edges is as follows: minimize c ii x ii i N {0} j N {0} j i, {i,j} C (3.12)

52 36 s.t.: j N {0} j i,i i N {0} i j,j x 0j = 1 j N (x ii + x i j ) = 1 {i, i } C (x ii + x ij ) = 1 {j, j } C (3.13) (3.14) (3.15) x j0 = 1 (3.16) j N t i + T ii t j +M ii 1 x ii i N, j N {0} i j, {i, j} C (3.17) a i t i i N {0} (3.18) t i b i i N (3.19) x ii x jj k N {0} k j i N {0}, j N {0} i j, {i, j} C (3.20) x ii {0,1} i N {0}, j N {0} i j, {i, j} C (3.21) t i R + i N {0} (3.22) The objective is to minimize the total traversal cost. Constraints (3.13) and (3.14) ensure that only one node is included in the tour for each pair of nodes that represent the same required edge. Constraints (3.15) and (3.16) force the tour to start and end at the depot. The time-window constraints are (3.17), (3.18), and (3.19). Constraints (3.20) guarantee flow conservation. Finally, the decision variables are defined in (3.21) and (3.22) Model on the nodes We now propose a transformation from the original problem to an equivalent problem on nodes. We define a new graph G 1 = (N 1, A 1 ), where all the vertices incident to the required edges

53 37 in the original graph G = (V, E) are included in the set of nodes N 1. It should be pointed out that if a vertex of V is incident to more than one required edge in G, then this vertex will have as many copies in N 1 as the number of incident required edges in G. A 1 is the set of arcs that connects the nodes of N 1. The cost of an arc that joins node i to node j is equal to the cost of the edge that starts at node i plus the length of the shortest path in G from the final vertex of that edge to the initial vertex j of the other edge. We set the cost to zero when nodes i and j represent vertices incident to the same edge in G or when i = j. We obtain a complete directed graph. We illustrate on an example the original graph in Figure 3.1a and the transformed graph in Figure 3.2 Note that there are two nodes labeled "4" and "4 " because they represent vertex "4" of Figure 3.1a, which is incident to two required edges Figure 3.2: Transformed graph for the model on the nodes We add an artificial node "0" to represent the depot. We join "0" to all nodes in N 1 by means of two arcs with costs equal to the length of the shortest path in G from the depot to the vertices, and from the vertices to the depot respectively. The time windows of each required edge in G are assigned to its incident nodes. In the transformed graph, we look for a minimum-cost tour that starts and ends at the depot and satisfies the time windows for all nodes. Additionally, two nodes incident to the same required edge must be placed one after the other in the tour sequence. Table 3.2 shows the cost matrix for the transformed graph.

54 38 Table 3.2: Costs of the transformed graph Model on the nodes To From Let us define C 0 as the set of pairs of nodes that are incident to the same required edge, and U as the set of nodes that includes one of the two nodes incident to the same required edge. Furthermore, let the parameters c ii, T ii, [a i b i ], a 0, and M ii and the decision variables x ii and t i be as defined in Section 2.2. The formulation on the nodes is as follows: minimize c ii x ii i N 1 {0} j N 1 {0} j i (3.23) s.t.: x ii + x i i = 1 {i, i } C 0 (3.24) x ii = 1 i N 1 {0} (3.25) j N 1 {0} j i x ii = 1 j N 1 {0} i N 1 {0} i j (3.26) t i + T ii t j +M ii 1 x ii i N 1, j N 1 {0} i j (3.27) a i t i i U {0} (3.28) t i b i i U (3.29) t i = t i {i, i } C 0 (3.30)

55 39 x ii {0,1} i N 1 {0}, j N 1 {0} i j (3.31) t i R + i N 1 {0} (3.32) The objective is to minimize the total traversal cost. Constraints (3.24) are related to the required services. Constraints (3.25) and (3.26) ensure that each node is visited. The timewindow constraints are (3.27), (3.28), (3.29), and (3.30). The decision variables are defined in (3.31) and (3.32). For the model on the edges we cannot associate a unique starting and completion time with each edge; we need the indices k and l to identify the number of times each edge would be traversed in a deadheading mode for the minimum distance objective. Therefore we must augment the initial graph k l times. The graph for models based on transformations includes only the incident nodes to required edges or the required edges. The other edges (not required) are considered only for getting the shortest path among required edges. Consequently, the variables do not need an extra index to identify the number of times edges are traversed in a deadheading mode (required edges are visited no more than once in the modified graph). The models based on the transformations have fewer variables and constraints than the model on the edges. They also use large integer values M ii, but these can be bounded to minimum values that allow us to find feasible solutions, therefore these formulations are better bounded. 3.3 Valid inequalities We focus on the model on the nodes (Section 3.2.3), taking advantage of its structure. This model has elements of the precedence constrained asymmetric TSP (PC-ATSP). Polyhedral approaches to solve problem instances to optimality are known to work well for the PC-ATSP (Ascheuer et al. 2000), as already mentioned, for the TSP. We study some of the known valid inequalities with respect to the two problems that are also valid for the formulation (3.23) (3.32). In the following, we summarize the classes of inequalities that we use in our solution method.

56 40 Notation Given the set of arcs A f, for any arc set W A f we define x(w) x ii {i, j} W). Given the set of nodes V f that includes the depot "0", for any two node sets S, T V f we define (S: T) {i, j} A f i S, j T and write x(s: T) for x (S: T). Lifted t-bounds. Desrochers and Laporte (1991) observed that the bounds of the t-variables (see inequalities 3.28 and 3.29) can be strengthened. Indeed, let a jj = max 0, a j a i + T jj and b ii = max 0, b i b j + T ii. Then the inequalities a i + n j=1 j i a jj x jj t i i V f {0} (3.33) b i n j=1 j i b ii x ii t i i V f {0} (3.34) are valid for the formulation (3.23) (3.32). Strengthened MTZ-inequalities. Desrochers and Laporte (1991) propose a lifted version of the MTZ subtour-elimination constraints (3.27). Let a jj = max T jj, a i b j and M ii b i + T ii a j. Then for all i, j = 1,, n, i j the inequality t i + T ii 1 x ii M ii + M ii T ii a jj x jj t j (3.35) is valid for the formulation (3.23) (3.32). According to Desrochers and Laporte (1991), when precedence relations exist, the MTZinequalities can be further strengthened. Assume i j. Since i must be scheduled before j, we have t i t j, and the inequality t i + T ii x ii t j (3.36) is also valid. If b i + T ii a j holds, inequality (3.36) can be strengthened to t i + T ii x ii a j (3.37) Subtour elimination constraints. We include the subtour elimination constraints, since they are the best known inequalities for the Asymmetric Traveling Salesman polytope (Balas et al. 1995). These inequalities x(s S) S 1 can be written in the equivalent cut form

57 41 x(s S ) 1 S, S V f (3.38) where S V f S, and (3.6) is valid for the formulation (3.23) (3.32). The Predecessor/Successor inequalities. The PC-ATSP is a relaxation of the ATSP-TW. We use some valid inequalities for the PC-ATSP that allow us to strengthen the subtour elimination inequalities (3.38). Balas et al.(1995) introduced these classes of inequalities. For S V f {0}, S V f S, the predecessor inequality (π-inequality) x S π(s) S π(s) 1 (3.39) and the successor inequality (σ-inequality) x S σ(s) S σ(s) 1 (3.40) are valid for the formulation (3.23) (3.32). For any given i, k V f {0} such that π(i), σ(k), and any S V f such that i, k S, the inequalities x S π(i) S π(i) 1 (3.41) x S σ(k) S σ(k) 1 (3.42) are called weak π- and weak σ-inequalities respectively. 3.4 Solution algorithm We implement the following algorithm to solve the Undirected RPPTW Data preprocessing Data preprocessing is important for efficient implementations. It allows the construction of tighter equivalent formulations of the problems, such that no optimal solution of the original problem is lost and each solution of the tighter problem corresponds to a solution of the original problem. The structures of the formulations on the required edges (Section 3.2.2) and on the nodes (Section 3.2.3) permit such a preprocessing procedure. It is based on the work of Ascheuer et al. (1999). We tighten the time windows iteratively until no more changes are made. We then

58 42 identify precedence relations, fix variables permanently, and detect infeasible paths of size two and three to reduce the set of variables. We now present the separation procedures for the classes of valid inequalities (3.38) (3.42) Cutting plane algorithm Initial linear program. We solve the relaxation of the model on nodes (3.23) (3.32), i.e., when the decision variables x ii are restricted to be nonnegative and less than or equal to one. Constraints (3.33) and (3.34) are included in the initial model instead of constraints (3.28) and (3.29) because the former are stronger. We also include the strengthened MTZ-inequalities (3.35), (3.36), and (3.37) instead of the MTZ-inequalities (3.38) when possible. Separation routines. Let (x, t ) be a solution where x is fractional. We want to identify a member of a family F of valid inequalities listed in Section 3.3 for the formulation on the nodes that is violated by x or else show that x satisfies all members of F. The implemented separation procedures are an adaptation of routines described in the literature. Subtour elimination constraints: For the cutset inequalities (3.38) we can solve the separation problem by computing the connected component T that includes the depot in the graph G induced by x ii > 0. If this component does not include all the nodes in V f, the subtour elimination constraint is violated by x, and we obtain the set S that includes all nodes in T. This procedure detects inequalities (3.38) which are violated only in the case where there is no path in G from the depot to any j V f {0}. Predecessor inequalities: We implement the exact separation algorithm presented by Balas et al.(1995) for the separation problem of predecessor inequalities. Although this algorithm detects only a violated weak π-inequality, if one exists, rather than a stronger π-inequality of the class (3.39), the detected violated inequality (3.41) can be replaced with a strictly stronger violated inequality of the class (3.39) when we include π(s) instead of π(j). If we apply the algorithm for j V f {0} such that π(j) =, we detect the known cutset inequality, and

59 43 obtain an algorithm that simultaneously solves the separation problem for both the subtour elimination inequalities and the π-inequalities. The successor inequalities: With a similar procedure to Balas et al.(1995) we can detect if x violates a weak σ-inequality. For any fixed j with σ(j), delete σ(j) from V f and in the resulting network with arc capacities x ii try to send one unit of flow from node 0 to node j. If this is possible, all inequalities (3.42) associated with the given j are satisfied by x ; otherwise the minimum capacity identified by the failed attempt to send a unit of flow specifies the σ-inequality most violated by x. We reverse the sets S and S, i.e., S is replaced by S and vice versa. As in the previous case, if a violated inequality (3.42) is found, it is replaced with a strictly stronger violated inequality of the class (3.40), when we include σ(s) instead of σ(j). Steps for the separation routine Subtour elimination constraint routine. Shrinking : The separation algorithm for the predecessor/successor inequalities implies the computation of the maximum flow for each pair i, j V f. We use shrinking procedures (Padberg and Rinaldi, 1990) to reduce the problem size and to avoid as many maximum-flow calculations as possible. Shrinking checks whether or not certain nodes lie on the same side of a minimum-capacity cut. If the results are positive, the subset is contracted or shrunk to a single node. We contract nodes i and j if they are incident to the same required edge. Also, we contract nodes i and j if x ii = 1 in the fractional solution x. Predecessor inequalities: We use the separation problem for the predecessor inequalities, and we simultaneously check the subtour elimination constraints when there is one connected component in the fractional solution x. Successor inequalities: We use the separation problem for the successor inequalities, and we simultaneously check the subtour elimination constraints when there is one connected component in the fractional solution x. We generate at most one cutting plane for each separation routine per iteration. The linear problems are solved using standard parameters of CPLEX

60 Solution of the MIP program We stop the cutting plane algorithm whenever the last 10 linear problems produce no improvement in the lower bound, or in case the improvement is less than 0.1%, or when the running time reaches three hours. In those scenarios the decision variables x ii are restricted to be binary, we add the valid cuts, if any and we solve the problem using the callable library of CPLEX with its default parameters except that the number of threads is set to Computational results In this section we describe the results of the comparison of the models on a set of randomly generated instances and the performance of our cutting plane algorithm which was tested also on a set of instances based on the real network of the Estrie region in Quebec. Our implementation is coded in Python 2.6 and runs on a 2.38 GHz AMD Generated instances There are no published benchmark instances for the undirected RPPTW. We modified the CARP-TW instances of Wøhlk (2005). The author combines five values for the number of nodes ({10,13,20,40,60} ) and the number of edges ({15,23,31,69,90}) and generates different graphs for each combination. We selected some required edges randomly and found a path through all of them using the nearest-neighbor heuristic. Then, we established the time-window intervals by reducing and extending by 10, 30, and 50% the values of the arrival time given by the heuristic. In this way we label the width of the time windows with {10,30,50}. When we combined the percentage of required edges {10,30,50} and the width of the time windows {10,30,50} we generated 225 instances. All the instances can be downloaded from Instances based on the Estrie network We tested the cutting plane algorithm on a real undirected network that represents a part of the Estrie administrative region in the province of Quebec, Canada. The network has 140 nodes and

61 edges. We simulated nine weather forecasts for one day with 4 or 5 time slots, each time slot defining the time windows. If any edge is located in a time slot with rain forecasted, the edge will be required to be visited in its respective time slot. Figure 3.3 shows an example of simulated weather forecasting for Estrie region for one time slot. The colors represent a different probability of rain. We define the cost of traversal as the length of the road multiplied by a fractional number in order to get a scalar representation, and the time of traversal proportional to this value. Figure 3.3: Estrie network Weather forecast for one time slot Preprocessing As noted earlier, the structures of the formulations on the required edges (Section 3.2.2) and on the nodes (Section 3.2.3) allow data preprocessing. Table 3.3 shows the effect of data preprocessing, giving the average percentage of removed variables. In general, there was a considerable reduction in the problem size. When the time windows are tighter more variables can be fixed.